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<b>let </b><font color="Maroon">G</font> be   <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">v</font> is <a href="chord.html#V3" target="_self">simplicial</a>
 ;<br/>

<b>assume </b><a NAME="E2:81"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>  holds <br/>(  not <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> or  not <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">a</font> or  not <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> or  not <font color="Maroon">v</font>,<font color="Olive">a</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  or  not <font color="Maroon">v</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  or <font color="Olive">a</font>,<font color="Olive">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1"/>
( <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><b>suppose </b></a><a NAME="E1:81_1_1"/>
<font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2"><b>suppose </b></a><a NAME="E1:81_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be    <a href="chord.html#M1" target="_self">AdjGraph</a> of <font color="Maroon">G</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E1:81_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">H</font> is   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>, <font color="Maroon">b</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">H</font>;<br/><b>assume </b><a NAME="E1:81_1_2_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:81_1_2_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="txt" href="chord.html#E5:81_1_2_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E3:81_1_2_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chord.html#K2" target="_self">.AdjacentSet</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">v</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><b>then reconsider </b><font color="Maroon">vv</font> = <font color="Maroon">v</font>, <font color="Maroon">aa</font> = <font color="Maroon">a</font>, <font color="Maroon">bb</font> = <font color="Maroon">b</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> ;<br/><a NAME="E5:81_1_2_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">aa</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">vv</font> &amp; <font color="Maroon">aa</font>,<font color="Maroon">vv</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E6:81_1_2_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">bb</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">vv</font> &amp; <font color="Maroon">bb</font>,<font color="Maroon">vv</font> <a href="chord.html#R1" target="_self">are_adjacent</a>  )
 <b>by </b><i>, <a class="ref" href="chord.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E7:81_1_2_1_1"/>
<font color="Maroon">aa</font>,<font color="Maroon">bb</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E2:81_1_2_1_1"><i><font color="Green">E67</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E8:81_1_2_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="chord.html#R1" target="_self">are_adjacent</a> 
 <b>by </b><i><a class="txt" href="chord.html#E5:81_1_2_1_1"><i><font color="Green">E68</font></i></a>, , <a class="ref" href="chord.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:81_1_2_1"/>
<font color="Maroon">H</font> is <a href="chord.html#V2" target="_self">complete</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E3:81_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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